#1. Experimentally, I'd say no. There's no such thing as an absolutely straight line in the real world. Therefore, you couldn't measure length or area exactly, because you'd have to assume straight lines somewhere, which don't exist. Any technique you could think of to measure volume would have inaccuracies.

Theoretically, you also have problems: just think of length. As you use smaller and smaller ruler lengths (as in, microscopic), you can prove that the length goes to infinity (you're moving back and forth more and more). If you're going around molecules, and you run into quantum mechanics, you now have the additional difficulty of defining length in the first place.

#2: Again, I would say no, for reasons I've outlined above.

The branches of mathematics to which this question belongs would include fractals, quantum mechanics, and experimental science.

Thank you Adrian. It's most valuable to know that there indeed is a sound logical basis for this question - i.e. that it isn't trivial. However, I'm hoping to find a more rigorous treatment of the problem; perhaps a theorem or generally accepted conjecture that might apply.

My literature search on this question has thus far been mainly in the field of chaotic systems - in particular, the initial conditions for non-linear systems, where minor differences in physical dimension matter greatly. But I've found the discussion revolves principally around measurement 'errors' (that compound over time or over iterations), as opposed to actual differences of sub-microscopic scale in the spatial dimensions of physical bodies. Specifically, the model I'm working on requires such differences to exist in the 'volumes' of any two like bodies.

In quantum mechanics, the relevant area presumably would be the Heisenberg uncertainty relationships. However, since these are very specific to a quantified inequality in the inverse relationship between wave-particle momentum and position, I don't know of a way that the underlying principles might be applied to the question of physical dimension differences.

My (limited) understanding of Mandelbrot fractals is that the self-similarity at successive magnification scales - and resulting infinite boundary length - is a consequence of the iterations performed on the complex polynomial; rather than something inherent in respect of the degree of precision of a length's measure. But I may well be wrong in this!

I'd be grateful for any additional thoughts, or references, on finding a proof for the proposition that: "the volumes of any two like bodies cannot be equal to an unlimited degree of precision (or that the probability of this being so approaches 1)".

Ackbeet said "experimentally". You cannot actually measure anything to arbitrary precision because of limitations in measuring tools.

Mathematically, there is no reason why we cannot measure to an "arbitrary precision". Since "self-similarity", fractals, and Mandelbrot and Julia sets are mathematical concepts, the impossibility of actually measuring to arbitrary precision has nothing to do with them.

And there cannot be a proof of the mathematical assertion that "The volumes of an two like bodies cannot be equal to an unlimited degree of precision".

My argument here is that assuming volumes are (theoretically) measurable with absolute precision, then within a set of like physical bodies, the probability of finding two or more bodies whose volumes are exactly equal (to unlimited precision) must be near-zero - though I accept that it cannot be zero.

The rationale is as follows: If we take one body from a set of like physical bodies (say, a set of 1,000,000 finely crafted billiard balls) as the measurement standard, we would find that its volume differs, at some digit after the decimal, from approximately all others in the set. Since the theoretical possibility cannot be ruled out that at least one body within the set is of a volume exactly (i.e. to infinite decimal digits) equal to that of the standard body, the probability cannot be zero.

How do I go about formulating/tackling this? Could countability of irrationals vs. rationals be of use here, and how?

I think that, at this level, you're going to have to deal at the particle level. In order for two different physical bodies to have the same exact mass, there are a number of things that would have to be in place (and this is assuming, as you have indicated, that it actually is theoretically possible to measure the masses to any arbitrary precision):

1. The two bodies would have to have precisely the same configuration of the same particles in every respect.
2. If the energies differ by any amount, then relativity theory says you've got different masses. And since the energies are quantized, they must be precisely the same.

There may be other restrictions. But these constraints, while theoretically satisfiable, I suppose, are, of course, highly unrealistic in the laboratory.

You're dealing with a problem in statistical physics. You have an ensemble, a regular pattern of particles. Each micro-state of each particle is equally likely. You have on the order of Avagadro's number of particles, most likely, perhaps orders of magnitude more. From there you can start to compute. The probably of this happening certainly does approach zero extraordinarily quickly. But you're right in that the probability is not, I think, identically zero.

Actually the problem is not with mass but with 'volume'. As volume is a continuous variable, its value for any given physical (i.e. real, material) body would differ at some digit after the decimal from all others in a group of like bodies with a near-1 probability.

I agree fully with your point about 'mass', for which values should be discrete due to energy quantization.

Please note that while the question is about physical bodies, precision in the measurement of their volumes is an "in principle" matter. Our practical ability to measure them to any arbitrary precision in the laboratory is therefore not of concern.

Also, it's only 'relative' values that matter. An illustrative thought experiment could be: If 10 like particles are placed randomly in a row and they somehow can "sense" the difference in their own volume relative to the others, then, if required to be next only to their nearest neighbors, the particles would self-organize in sequential order of volume, irrespective of the minuteness of the differences.

Considering the continuous nature of the object variable, does it remain a problem of statistical physics?

Well, the original problem is about volume, but I think the problem is going to have to reduce down to one of mass. The reasoning might go like this: A. Bodies made up of different elements almost certainly cannot have the same volume, because the geometry of the solid will likely be different. B. If the two bodies are made of the same elements, then you're likely going to want to have them in precisely the same spatial configuration (perhaps at standard temperature and pressure). If that's the case, then identical masses would, theoretically, mean identical volumes. I realize this reasoning is not bi-directional.

And your thought about the volumes differing at some digit is appropriate, I think. If one electron has an energy level different from one body to another, then you wouldn't have the same volume.

Well, the original problem is about volume, but I think the problem is going to have to reduce down to one of mass. The reasoning might go like this: A. Bodies made up of different elements almost certainly cannot have the same volume, because the geometry of the solid will likely be different. B. If the two bodies are made of the same elements, then you're likely going to want to have them in precisely the same spatial configuration (perhaps at standard temperature and pressure). If that's the case, then identical masses would, theoretically, mean identical volumes. I realize this reasoning is not bi-directional.

And your thought about the volumes differing at some digit is appropriate, I think. If one electron has an energy level different from one body to another, then you wouldn't have the same volume.

The problem I'm dealing with is actually about sub-atomic particle 'densities'. So a particle's 'mass' and 'volume' require separate treatment.

Since the problem concerns individual particles only, rather than composites such as molecules or macroscopic bodies, spatial geometry is not a factor.

NOTE: Quantum mechanical ambiguities about the physical character of a particle are ignored, whereby a particle is treated simply as a material, but elastic (i.e. not rigid), body. Moreover, in this (hypothetical) framework, changes in a particle's mass (or equivalently in its energy content) do not have a one-to-one correspondence with changes in its volume - volume can vary also for numerous environmental reasons unrelated to any mass-energy changes.

In short, while a particle's mass and volume are related (corresponding to volumetric thermal expansivity in macroscopic bodies), variability in a particle’s volume is not solely a function of changes in its mass-energy.

Mass can be identical for two or more particles due to energy quantization. But volume, as a continuum variable, can have any non-zero (rational or irrational) positive value. So density also can have any non-zero (rational or irrational) positive value under this framework.

OBJECTIVE: The aim is to establish in principle that, given such a hypothetical framework, there is a near-zero probability of finding two particles with densities that are exactly equal (i.e. to a precision extending to infinite decimal digits). This does not in any way preclude the possibility of the densities of all or many particles being extremely close (say, to 10^10th or 10^50th decimal digits or greater).

For perspective, an analogous statement would be that: There is a near-zero probability of finding two leaves in the Amazon that are of exactly equal volume.

I see. Everything is at the particle level. Does that mean you're looking at atoms? Or protons? Electrons? How fundamental are your particles? You are allowing mass/energy to be quantized, but not volume. That seems a bit arbitrary to me. I realize you're trying to avoid quantum mechanics when it comes to volume. However, if you fix the mass/energy, then the volume is determined! (Well, the wave function is determined, which will tell you the probability of finding elements of mass within a particular volume element). If you think about an electron "orbiting" a proton, if you determine the energy of the electron, then you know how far away it is from the proton (Bohr radii), while the probability of finding the electron within a particular area of its orbit is given by the integral of the wave function modulus squared over that area.

If you still want to take volume as continuous, then your probabilities are definitely going to approach zero.

I see. Everything is at the particle level. Does that mean you're looking at atoms? Or protons? Electrons? How fundamental are your particles? You are allowing mass/energy to be quantized, but not volume. That seems a bit arbitrary to me. I realize you're trying to avoid quantum mechanics when it comes to volume. However, if you fix the mass/energy, then the volume is determined! (Well, the wave function is determined, which will tell you the probability of finding elements of mass within a particular volume element). If you think about an electron "orbiting" a proton, if you determine the energy of the electron, then you know how far away it is from the proton (Bohr radii), while the probability of finding the electron within a particular area of its orbit is given by the integral of the wave function modulus squared over that area.

If you still want to take volume as continuous, then your probabilities are definitely going to approach zero.